There is a fundamentally incorrect assumption with your question, “—…and I note that they have "Questioners", whose entire job, as far as I can tell, is to root out hidden Aes Sedai. But Aes Sedai are sworn never to lie… In the books, is there a reason questioners don't just do this to find them?”
The job of the “Questioners,” formally known as the Hand of Light “inquisitor” sect are part of the Children of Light organization which has made its job to root out Darkfriends — humans who have been tempted with promises of immortality and power by the Dark One, or by other Darkfriends, into serving the Shadow. The “Questioners” are led by the High Inquisitor, and seek to secure confessions of alleged Darkfriends primarily through brutal torture. The sticking point with your question is that the Children of Light view all Aes Sedai as Darkfriends; this belief then is basis for your question whether the point is get all Aes Sedai as opposed to get Darkfriends. Out-of-universe, and unfortunately for innocent Aes Sedai, they are wrong in their belief, and this conflict serves as part of the plot narrative of the Wheel of Time saga.
Darkfriends can especially include Aes Sedai. Aes Sedai who are Darkfriends are members of the Black Ajah, a secret Ajah, consisting of Darkfriend Aes Sedai who masquerade as sisters of other Ajahs and who have forsaken their Three Oaths in order to serve the Dark One. The Questioners haven’t bought into such a distinction, however.
Because Black Ajah lie, the Hand of Light (“Questioners”) take special interest in interrogating Aes Sedai of any Ajah, on the possibility they may be Black Ajah. This inquisition of the Aes Sedai also serves the Questioners’ belief that all Aes Sedai are Darkfriends because their brutal methods generally yield confessions (then death), or death regardless of whether the victim was a Darkfriend or not.
The Hand of Light is an allegorical representation of Spanish Inquisitors, and the brutal methods of interrogation.
In a Q/A chat, Robert Jordan has confirmed all of this that the Whitecloaks started as a religious sect, evolved into a military organization in the War of the Hundred Years, who exhibit a touch of the Spanish Inquisition:
WOOD SUN And how about the Whitecloaks? I mean they look like some sort of religious sect.
ROBERT JORDAN Which?
QUESTION (two girls in unison) The Whitecloaks!
ROBERT JORDAN The Whitecloaks? Well, they're meant to look as a religious sect. They began as, an ascetic organization dedicated to preaching against Darkfriends, trying to convince people by example that they should not become Darkfriends. And during the War of the Hundred Years they became a military organization. They are patterned on the Teutonic Knights, a touch of the German SS, and...
WOOD SUN [interrupts] And the Spanish Inquisition?
ROBERT JORDAN A touch of the Spanish Inquisition. (laughter) They are in short anyone who believes that they know the Truth—the Truth with a capital T. They know the Truth so well, and its so clear to them that if you don't believe that truth, then it becomes obvious that you are evil.
The Eye of the World quickly sets this tone, and Chapters 29 and 30 are especially notable for the “good cop, bad cop” routine of Whitecloaks Bornhald and Byar, which is actually “bad cop, worse cop.” Bornhald states his belief that Egwene and Perrin might be lead away from the Shadow and brought back to the light; or Egwene, at least. For Perrin, who killed two of the Children of the Light in his rage over Hopper’s death, he says there can only be one fate:
Neither Bornhald nor Byar have proof of any of the characters being Darkfriends, but in case of Children of Light, it’s guilty until proven innocent — a belief that falls particularly hard on Aes Sedai.